The complete symmetry group of the generalised hyperladder problem

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J. Math. Anal. Appl. 293 (2004) 633–644

www.elsevier.com/locate/jma

The complete symmetry group of the generalisehyperladder problem

K. Andriopoulos1 and P.G.L. Leach∗,2

Research Group for Mathematical Physics, Department of Information and Communication SystemsEngineering, University of the Aegean, 83 200 Karlovassi, Greece

Received 3 January 2004

Submitted by K.A. Ames

Abstract

We further consider then-dimensional ladder system, that is the homogeneous quadratic sysfirst-order differential equations of the formxi = xi

∑nj=1 aij xj , i = 1, n, where(aij ) = (i +1− j),

i, j = 1, n introduced by Imai and Hirata (nlin.SI/0212007). We establish the most general sysfirst-order ordinary differential equations invariant under the algebra which characterises thesystem of Imai and Hirata and the algebra of minimal dimension required to specify compthis most general system. We provide the complete symmetry group of the generalised hypesystem and discuss its integrability. 2004 Elsevier Inc. All rights reserved.

1. Ladder systems

In two recent papers Imai and Hirata developed a necessary condition for the existeof Lie point symmetries inn-dimensional systems of first-order ordinary differential eqtions [5] and applied the ideas developed there to establish a new integrable familyclass of Lotka–Volterra systems [6]. Of the infinite number of Lie symmetries that s

* Corresponding author.E-mail addresses:[email protected] (K. Andriopoulos), [email protected], [email protected]

(P.G.L. Leach).1 Permanent address: Department ofMathematics, National and Capodistrian University of Athens

Panepistimioupolis, Ilisia, 15771 Athens, Greece.2 Permanent address: School of Mathematical and Statistical Sciences, University of Natal, Durban 404

Republic of South Africa.

0022-247X/$ – see front matter 2004 Elsevier Inc. All rights reserved.doi:10.1016/j.jmaa.2004.01.023

634 K. Andriopoulos, P.G.L. Leach / J. Math. Anal. Appl. 293 (2004) 633–644

s of the

thatof

ay

truc-and

raintsy the

duc-

trix

y

of thethe

sys-ence

system possesses Imai and Hirata [5] were concerned with autonomous symmetrieform

Γ = φi(x)∂xi (1)

for the autonomous system

x = g(x), (2)

whereg(x) is analytic in the neighbourhood of a fixed point. Moreover the symmetry,is to say the coefficient functionsφi , was required to be analytic in the neighbourhoodsome fixed point.

In their application of the results of [5] ton-dimensional homogeneous, that is to squadratic, Lotka–Volterra systems of the form

xi = xi

n∑j=1

aij xj , i = 1, n, (3)

which arise in a variety of applications, Imai and Hirata [6] considered two specific stures for then-dimensional homogeneous Lotka–Volterra system, the ladder systemthe generalised ladder system. The former is characterised by the possession of an(n− 1)-dimensional Abelian Lie algebra of analytic symmetries of the type (1). The constof the latter so that it is integrable in terms of analytic functions are determined bimposition of a like Abelian algebra.

The integrability of the ladder system in terms of the Painlevé analysis, explicit retion and generation of solutions was given recently [3].

Then-dimensional ladder system is the system (3) with the specific coefficient ma

A = (aij ) =

1 0 . . . −n + 22 1 . . . −n + 3...

.... . .

...

n n − 1 . . . 1

, (4)

where

aij = i + 1− j, (5)

ai+1,j − aij = 1. (6)

The elements of the coefficient matrix of the generalised ladder problem are given b

aij = 1+ ai − aj . (7)

The ladder system has certain general properties [3,5,6]. According to [3] the rankmatrix A is two for n > 2 and then-dimensional ladder system can be considered assimplest Riccati equation for

∑ni=1 xi and as such is an example of a decomposed

tem [4]. Furthermore then-dimensional ladder system is consistent at all resonances, hit passes the Painlevé test. Finally the general solution of then-dimensional ladder systemis given by

xi+1 = x1

∏ij=1 Kj

i,

(t − t0)

K. Andriopoulos, P.G.L. Leach / J. Math. Anal. Appl. 293 (2004) 633–644 635

eyla

s ane lad-itly [3].s

rily

umberatter

alstem.Sub-1].ationsystems

find

l than

. How-adder

alsoveof our

where

x1 = − (t − t0)n−2∑n−1

j=0(∏j

k=0 Kk)(t − t0)n−1−j.

Imai and Hirata [6] presented then2 − 1 set of symmetries

Y lm = xm

u1+m−l

(n∑

j=1

xj ∂xj − u∂xl

), l,m = 1, . . . , n, (8)

whereu = x1 + · · · + xn, for the system (3) with the specific coefficient matrix (6). Thwere particularly interested in the symmetriesYn

m, m = 1, n − 1 which have polynomiacoefficient functions and hence have the property of being analytic symmetries. This wasfeature of relevance to their earlier work [5]. This particular set of symmetries possesseAbelian algebra and from this property Imai and Hirata were able to conclude that thder system was integrable. This conclusion was subsequently demonstrated explicThe generalised ladder problem with the coefficient matrix (7) possesses the symmetrie

Y lm = xm

u1+am−al

(n∑

j=1

xj ∂xj − u∂xl

). (9)

Imai and Hirata [6] concluded that the system was also integrable although not necessain terms of analytic functions.

The very integrability of the ladder system and its possession of a generous nof symmetries of this specific form prompts a number of questions. The first is a mof uniqueness. We recall that the complete symmetry group of a system of differentiequations is the minimal number of symmetries required to specify completely the syThe concept was introduced by Krause [7,8] in the context of the Kepler problem.sequently the theoretical basis and applications have received some attention [1,2,9–1In this respect we must observe that systems of first-order ordinary differential equhave been generally not considered. The main thrust of the attention has been to sof second-order ordinary differential equations.

In this paper we determine the most general form of a system ofn-dimensional first-order ordinary differential equations invariant under the symmetries given in (9). Weit to be

xi =[xi

n∑j=1

(1+ ai − aj )xj

]φ(t, u), (10)

whereφ is an arbitrary function of its arguments, which is somewhat more generathe homogeneous Lotka–Volterra system treated by Imai and Hirata. Then2 − 1 linearlyindependent symmetries in (9) are more than is necessary to specify completely (10)ever, we must add additional symmetries, not listed in (9), to specify completely the lsystem of Imai and Hirata. The integrability of our more general system is moot. Wepresent some features of the algebraic structure of the symmetries given in (9) which hanot previously been reported. We conclude with a summary of the specialisationresults to the generalised ladder problem presented by Imai and Hirata [6].


2. Complete symmetry groups of ladder problems

Certain properties of the symmetries listed in (9) are needed below. They are [6][Γ l

m,Γ l′m′

] = δm′,lΓl′m − δm,l′Γ

lm′ , (11)

n∑l=1

Γ ll = 0, (12)

Γ lm(u) = 0, (13)[Γ n

1 ,Γ nn−1

] = 0. (14)

Proposition 1. The2n − 2 symmetries

Ynm = xm

u1+am−an

(n∑

j=1

xj∂xj − u∂xn

), m = 1, n, (15)

Y ll = xl

u

(n∑

j=1

xj∂xj − u∂xl

), l = 2, n − 1, (16)

specify the generalised hyperladder system

xi =[xi

n∑j=1

(1+ ai − aj )xj

]φ(t, u), i = 1, n.

Proof. Consider the system

xi = fi(t,x), i = 1, n. (17)

The first extension ofYnm is

Ynm

[1] = Ynm +

(fm

u1+am−an− (1+ am − an)xmf

u2+am−an

)(n∑

j=1

xj ∂xj − u∂xn

)

+ xm

u1+am−an

(n∑

j=1

fj ∂xj − f ∂xn

), m = 1, n,

wheref = ∑nj=1 fj . The action ofYn

m[1] on the system (17) gives(

fm

xm

− (1+ am − an)f

u

)(xi − uδin) + fi − f δin

=n∑

j=1

xj∂fi

∂xj

− u∂fi

∂xn

, i,m = 1, n.

In the case thati = 1 andn > 1, that is the nontrivial case, we have(fm

xm

− (1+ am − an)f

u

)x1 + f1 =

n∑xj

∂f1

∂xj

− u∂f1

∂xn

, m = 1, n. (18)
j=1


mes

a-

n as

Definefm = xmφm, m = 1, n, and no summation on the repeated index. Then (18) beco

uφm − (1+ am − an)

n∑j=1

xjφj = u

n∑j=1

xj∂φ1

∂xj

− u∂φ1

∂xn

, m = 1, n. (19)

We observe that the right side of (19) is the same for all values ofm. Thus we equatesuccessive left sides of (19) to obtain

u(φm+1 − φm) = (am+1 − am)

n∑j=1

xjφj , m = 1, n − 1. (20)

We assume thatam+1 �= am (the caseam+1 = am is considered below). Then the combintions

(am+2 − am+1)(20)(m,m+1) − (am+1 − am)(20)(m+1,m+2)

give

(am+2 − am+1)(φm+1 − φm) = (am+1 − am)(φm+2 − φm+1)

whence

φm+2 = 1

am+1 − am

[(am+2 − am)φm+1 − (am+2 − am+1)φm

]⇒ φj = aj − a1

a2 − a1φ2 − aj − a2

a2 − a1φ1, j = 3, n. (21)

We substitute (21) into (20) withm = 1 to obtain

u(φ2 − φ1) =n∑

j=1

xj (aj − a1)φ2 −n∑

j=1

xj (aj − a2)φ1

whence

φ2 = (1+ a2)u − ∑nj=1 ajxj

(1+ a1)u − ∑nj=1 ajxj

φ1 ⇒ φj = (1+ aj )u − ∑nj=1 ajxj

(1+ a1)u − ∑nj=1 ajxj

φ1. (22)

Equation (19) becomesn∑

j=1

xj∂φ1

∂xj

− u∂φ1

∂xn

= anu − ∑nk=1 akxk

(1+ a1)u − ∑nk=1 akxk

φ1. (23)

The associated Lagrange’s system for (23) is

dt

0= dx1

x1= · · · = dxn−1

xn−1= dxn

xn − u= dφ1

anu−∑nk=1 akxk

(1+a1)u−∑nk=1 akxk

φ1

. (24)

The characteristics from the firstn + 1 terms of (24) aret , u = ∑nj=1 xj andvj = xj/x1,

j = 2, n − 1. In terms of these characteristics the final equation of (24) can be writte

0 = dφ1

φ1−

∑n−1k=1(an − ak)vk

(1+ a − a )u + ∑n−1(a − a )v x

dx1,
1 n k=1 n k k 1


to

which implies that the final characteristic is

w = φ1

(1+ a1 − an)u + ∑n−1k=1(an − ak)vkx1

.

Hence

φ1 =(

n∑k=1

(1+ a1 − ak)xk

)φ, (25)

whereφ = φ(t, u,v) is an arbitrary function.Relation (22) gives the general result that

φi =(

(1+ ai)u −n∑

k=1

akxk

)φ =

(n∑

j=1

(1+ ai − aj )xj

)φ(t, u,v). (26)

The system (17) becomes

xi =[xi

n∑j=1

(1+ ai − aj )xj

]φ(t, u,v), i = 1, n. (27)

We now consider the action ofY ll for l = 2, n − 1 on (27),

1

uφ

{uxi

[(1+ al)u −

n∑j=1

ajxj

]− u2δil

[(1+ al)u −

n∑j=1

ajxj

]

− u2xi + xiu

[(1+ ai)u −

n∑j=1

ajxj

]}

=[(xi − uδil)(1+ ai)u

2 − (xi − uδil)u

n∑j=1

ajxj

− uxi

n∑j=1

ajxj + u2xial

]φ + xi

[(1+ ai)u −

n∑j=1

ajxj

]

×{

n∑j=1

xj

n−1∑k=1

∂φ

∂vk

(δkj

x1− xnδ1j

x21

)− u

n−1∑k=1

∂φ

∂vk

(δkl

x1− xkδil

x21

)}

in which we have used the property thatY lm[u] = 0.

The coefficient ofφ in the above equation vanishes. The remaining terms simplify

∂φ

∂vl

− δ1l

n−1∑k=1

vk∂φ

∂vk

= 0. (28)

Sincel = 2, n − 1,

(28) ⇒ ∂φ = 0.
∂vl


ies.)

ss

is

Thus

xi =[xi

n∑j=1

(1+ ai − aj )xj

]φ(t, u).

In the case wheream�+1 = am� we have from (20),

φm�+1 = φm� for 1 � m� � n − 1.

For 3� j � m�, (21) and (22) continue to hold. Form� +1� j � n−1, Eq. (20) continuesto hold. We assume, for the purposes of simplicity, thatam+1 − am �= 0, m� + 1 � m �n − 1. (If there exists such anm, then a repetition of the same argument below applForm = m� + 1 the relevant term in (20) is

u(φm�+2 − φm�+1) = (am�+2 − am�+1)

n∑j=1

xjφj

and the combination of this with (20)(1,2) gives

(a2 − a1)(φm�+2 − φm�+1) = (am�+2 − am�+1)(φ2 − φ1).

Since

φm�+1 = φm� = am� − a1

a2 − a1φ2 − am� − a2

a2 − a1φ1

⇒ φm�+2 = am�+2 − a1

a2 − a1φ2 − am�+2 − a2

a2 − a1φ1

and the general discussion can continue.�Note. The complete symmetry group found in Proposition 1 specifies completely the claof equations which we have called the generalised hyperladder system.

Remark. We have used the symmetriesYnm, m = 1, n, andY l

l , l = 2, n−1, for this demon-stration. One can equally useYm1

m2 , m2 = 1, n and 1� m1 � n − 1, andY ll , l = 1,m1 − 1,

m1 + 1, n − 1.

Proposition 2. The generalised ladder problem is specified by the2n − 2 symmetries(15)and(16)plus

Γ1 = ∂t , (29)

Γ2 =n∑

j=1

xj ∂xj − t∂t . (30)

Proof. The application ofΓ1 andΓ2 in turn to the system (10) requires thatφ(t, u) beconstant. �Proposition 3. The complete symmetry group of the generalised hyperladder problem

(n − 2)A1 ⊕s

{A1 ⊕s (n − 1)A1

}, (31)

wherejA1 is a j -dimensional Abelian group.


s

by ma-

Proof. From the property (11) we find that[Yn

m,Y nm′

]LB

= 0, m,m′ = 1, n − 1,[Y l

l , Yl′l′]LB

= 0, l, l′ = 2, n − 1,[Yn

m,Y ll

]LB

= −δmlYnl , m = 1, n − 1, l = 2, n − 1,[

Ynm,Y n

n

]LB

= Ynm, m = 1, n − 1.

Hence the result. �Corollary. The complete symmetry group of the hyperladder system is(31) with the sym-metries(15)and (16)with am andal replaced by the integersm andl.

Proposition 4. The complete symmetry group of the generalised ladder problem is

A1 ⊕s

{A1 ⊕ {

(n − 2)A1 ⊕s

{A1 ⊕s (n − 1)A1

}}}. (32)

Proof. The Lie brackets ofΓ1 andΓ2 are[Y l

m,Γ1]LB

= 0, m, l = 1, n,

[Γ1,Γ2]LB = Γ1,[Y l

m,Γ2]LB

= (am − al)Ylm, m, l = 1, n.

Hence the result. �Corollary. The complete symmetry group of the ladder system is(32)with the symmetrie(15), (16), (29)and(30)with am andal replaced by the integersm and l.

3. Integrability of the generalised hyperladder system

The first integrals of the generalised hyperladder system are easily constructednipulation of the system (10). Denoting theith member of the system byEi we see that

Em

xm

− El

xl

⇒ xm

xm

− xl

xl

= (am − al)uφ (33)

andn∑

i=1

Ei ⇒ u = u2φ (34)

so that (33) becomes

xm

xm

− xl

xl

= (am − al)u

u⇒ I l

m = xm

uam

ual

xl

. (35)

Equally one can follow Proposition 4 of [6] and note that, since


m the

rtial

Y lm′ = xm′

u1+am′−al

(n∑

j=1

xj ∂xj − u∂xl

)= xm′

xm

uam

uam′xm

u1+am−al

(n∑

j=1

xj ∂xj − u∂xl

)

= xm′

uam′uam

xm

Y lm, (36)

the function

Im′,m = xm′

uam′uam

xm

is a first integral as the coefficient ofY lm in (36) is autonomous.

Yet again one can use the standard method to construct the integrals directly frosymmetries, i.e., by determining the functionsI satisfying the dual conditions

Y lmI = 0 and I = 0. (37)

For notational convenience we make use of the symmetryYnm. The first of (37) gives the

linear homogeneous partial differential equationn∑

j=1

xj∂I

∂xj

− u∂I

∂xn

= 0 (38)

which has the associated Lagrange’s system

dt

0= dx1

x1= dx2

x2= · · · = dxn−1

xn−1= dxn

xn − u, (39)

n characteristics of which are

t, u andvj = xj

x1, j = 2, n − 1, (40)

whence

I = I (t, u,v). (41)

We apply the second of (37) toI now as in (41) to obtain the linear homogeneous padifferential equation

∂I

∂t+ u

∂I

∂u+

n−1∑j=2

vj

∂I

∂vj

= 0 (42)

with the associated Lagrange’s system

dt

1= du

u= dvj

vj

(43)

in which the final term representsn − 2 terms. We already haveu in (34). We calculatevj

as

vj = xj

x1− xj x1

x21

= 1

x1

{xj

n∑k=1

(1+ aj − ak)xkφ − xj

n∑k=1

(1+ a1 − ak)xkφ

}

= (aj − a1)vj uφ (44)


in

f theised

rdinaryally

tryis

t of the

sented

sitions

so that (43) is

dt

1= du

u2φ= dvj

(aj − a1)vjuφ. (45)

From the second and third of (45) we obtain then − 2 characteristics

wj1 = vj

uaj −a1= xj

uaj

ua1

x1, j = 2, n − 1,

each of which is a first integral. By a different choice of symmetry and denominatorvj

we recover all of the first integrals given in (35).Finally we have the first and second of (45) to consider. Without any knowledge o

specific form ofφ(t, u) it is not possible to find the required invariant. The generalhyperladder system—equally hyperladder since the elements of the matrixA are not thecause of the problem—cannot be reduced to a quadrature, but only to a first-order odifferential equation, in fact (34). As such the generalised hyperladder system is formintegrable, but not explicitly so as in the case of the generalised ladder system.

To be able to reduce the generalised hyperladder system to a quadrature a symmeother than theY l

m must be known. TheY lm simply give the first integrals of (35). Th

additional symmetry places a constrainton the occurrence of the variables withinφ(t, u).The generalised ladder system possesses two additional Lie point symmetries no

form Y lm. They are theΓ1 andΓ2 of Proposition 2 above. If we imposeΓ1 on (34), we

obtainφ = φ(u) and the invariant is

w1 = t −∫

du

u2φ(u)(46)

and we have reduction to a quadrature. If we imposeΓ2 on (34), we obtainφ = φ(ζ ),whereζ = ut . The solution of (34) is reduced to the quadrature

w2 = logt −∫

dζ

ζ(ζφ + 1). (47)

4. Discussion

The ladder systems introduced by Imai and Hirata [6] and the hyperladders prehere are characterised by the set of symmetries

Y lm = xm

u1+am−al

(n∑

j=1

xj ∂xj − u∂xl

), l,m = 1, n,

with the addition of

Γ1 = ∂t and Γ2 =n∑

j=1

xj ∂xj − t∂t

for the former systems. All of these ladder problems can be regarded as decompoaccording to definite rules of the single first-order ordinary differential equation

u = u2φ(t, u).


e

the gen-notl

o toditional

der and

eralisedn.

, as inof

leteropo-

m hash can

os-ersitynji

ns

Imai and Hirata [6] showed thatthe generalised ladder system

xi = xi

n∑j=1

(1+ ai − aj )xj , i = 1, n,

is integrable through the possession of an(n − 1)-dimensional Abelian algebra and thpossession of the first integrals

Ilm = xl

ual

uam

xm

, l,m = 1, n.

Here we have seen that the generalised hyperladder system

xi =[xi

n∑j=1

(1+ ai − aj )xj

]φ(t, u), i = 1, n,

in which φ(t, u) is an arbitrary function of both of its arguments, also has an(n − 1)-dimensional Abelian subalgebra and possesses the same set of first integrals aseralised ladder problem. For generalφ(t, u) the generalised hyperladder system isintegrable since the composed equation,u = u2φ(t, u), is not integrable. An additionasymmetry is required to enable this equation to be reduced to variables separable and sa quadrature. The generalised ladder system of Imai and Hirata possessed two adsymmetries.

We have presented the complete symmetry groups of both generalised hyperladladder systems. The former is 2(n− 1)-dimensional and the latter is 2n-dimensional. Bothare based on the possession of substantial Abelian subgroups. An integrable genhyperladder system requires a(2n − 1)-dimensional group for its complete specificatioThe relation of this subgroup to the rest of the group may be as a direct productthe case of the subgroup corresponding toΓ1, or a semidirect product, as in the casethe subgroup corresponding toΓ2. Finally we note that the representation of the compsymmetry group is not unique for either set of problems (see the Remark following Psition 1).

To our knowledge the complete symmetry group of a general first-order systenot previously been reported. It would be of interest to establish the subgroup whicspecify completely other systems of first-order ordinary differential equations.

Acknowledgments

K.A. and P.G.L.L. thank the University of the Aegean, in particular Professor G.P. Flessas, for their kind hpitality and provision of facilities during the Winter of 2003–2004 when this paper was written and the Univof Natal and the National Research Foundation of South Africa for their continuing support. We also thank KeImai for the provision of Ref. [5].

References

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[3] K. Andriopoulos, P.G.L. Leach, M.C. Nucci, The ladder problem: Painlevé integrability and explicit solutioJ. Phys. A 36 (2003) 11257–11265.

[4] K. Andriopoulos, P.G.L. Leach, Decomposition of the scalar autonomous Riccati equation, GEODYSchool of Sciences, University of the Aegean, Karlovassi 83 200, Greece, Preprint, 2003.

[5] Y. Hirata, K. Imai, A necessary condition for existence of Lie symmetries in general systems of ordifferential equations, J. Phys. Soc. Japan 71 (2002) 2396–2400.

[6] K. Imai, Y. Hirata, New integrable family in then-dimensional homogeneous Lotka–Volterra systems wAbelian Lie algebra, nlin.SI/0212007, 2002.

[7] J. Krause, On the complete symmetry group of the classical Kepler system, J. Math. Phys. 35 (19945748.

[8] J. Krause, On the complete symmetry group of the Kepler problem, in: A. Arima (Ed.), Proceedinthe XXth International Colloquiumon Group Theoretical Methods in Physics, World Scientific, Singapore1995, pp. 286–290.

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[10] P.G.L. Leach, Complete symmetry groups, the Kepler problem and generalisations, South African J. Sci(2003) 209–214.

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The complete symmetry group of the generalised hyperladder problem

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